Majorana boundary modes in Josephson junctions arrays with ...

Majorana boundary modes in Josephson junctions arrays with gapless bulk excitations M. Pino,1 A. M. Tsvelik,2 and L. B. Ioffe1, 3

arXiv:1505.01514v1 [cond-mat.mes-hall] 6 May 2015

1

Department of Physics and Astronomy, Rutgers The State University of New Jersey, 136 Frelinghuysen rd, Piscataway, 08854 New Jersey, USA 2 Department of Condensed Matter Physics and Materials Science, Brookhaven National Laboratory, Upton, New York 11973-5000, USA 3 CNRS, UMR 7589, LPTHE, F-75005, Paris, France.

The search for Majorana bound states in solid-state physics has been limited to materials which display a gap in their bulk spectrum. We show that such unpaired states appear in certain quasione-dimensional Josephson junctions arrays with gapless bulk excitations. The bulk modes mediate a coupling between Majorana bound states via the Ruderman-Kittel-Yosida-Kasuya mechanism. As a consequence, the lowest energy doublet acquires a finite energy difference. For realistic set of parameters this energy splitting remains much smaller than the energy of the bulk eigenstates even for short chains of length L ∼ 10.

Introduction. An intensive search for Majorana fermions [1] is underway in solid-state devices [2]. The vast majority of the proposals consist in zero energy boundary modes in materials with a gaped bulk spectrum. A spin-less superconducting wire or topological insulator in two or three dimensions fall in this category [3–5]. We propose an alternative approach for the observation of Majorana fermions in Josephson Junctions Arrays (JJA). We will show that certain quasi-one-dimensional JJA can display Majorana zero modes at their boundaries. These modes are protected from mixing with higher energy excitations although bulk spectrum is not gapped. The existence of low energy Majorana could then be proved by spectroscopy [6, 7]. In this letter, we first explain the JJA system and how to model it with an Ising-like Hamiltonian. Then, a qualitative argument is employed to obtain the low-energy effective theory using Majorana boundary modes. Numerical results will confirm the validity of this effective theory and show that Majorana modes are indeed protected. Finally, we discuss problems that may arise in the experimental realization of our proposal. Experimental set-up We consider three identical ladders of Josephson junctions coupled together as in Fig. (1a) [8]. Each ladder has a unit cell with ”large” and ”small” junctions, Fig. (1b). Their corresponding Josephson energies are EJL and EJS , where EJL > EJS . The three ladders are closed at the ends by a junction with Josephson energy EJE . We assume that charging energies are much smaller than Josephson ones for all the junctions. All the closed circuit in the ladders are at full frustration, that is, they are threaded by a magnetic flux equal to half of the flux quantum. The two larger loops are threaded by magnetic fluxes ϕ1 and ϕ2 , respectively. For ladder p, the phase difference in the left vertical and horizontal red junctions in the loop k are denoted by φp (2k) and φp (2k + 1), respectively (see Fig. 1b) . At full frustration, the properties of one infinite chain are invariant under translation by one small closed loop

FIG. 1. (a) Three ladders of Josephson junctions coupled together. The two large loops are threaded by magnetic fluxes ϕ1 and ϕ2 . (b) Unit cell of one ladder. Josephson energy of ”large” junction (red) is larger than the one of a ”small” junction (green), EJL > EJS . All the cells are threaded by a magnetic flux equal to half of the flux quantum. (c) Loop at the left boundary (the one at right is symmetric). The Josephson coupling for the blue junction at the chains end is denoted as EJE . (d) For small temperature and near a phase transition, the three Josephson ladders system maps to three Ising chains with transverse field (solid lines) and coupling between their ends (dotted lines). Majorana zero modes are located at the chains boundaries (dark regions).

and reflection through horizontal axes. We analyze the energy of one unit cell as a function of the phase differ-

2 ence in the ”large” junctions: n o Up,k = −EJL cos [φp (2k)] + cos [φp (2k + 1)] +EJS cos [φp (2k) + φp (2k + 1) + φp (2k + 2)] ,

(1)

where Up,k is the energy of the loop k of ladder p. The total energy has a global Z2 symmetry given by φp (i) = −φp (i) in each junction i. In the regime EJL  EJS , the ground-state corresponds to all phases equal to zero φp (i) = 0 and the symmetry is preserved. However, a broken symmetry phase occurs for EJL  EJS , as in this regime the ground-state acquires a φp (i) 6= 0. The critical point is located near EJL /EJS ∼ 5 [8]. The specific details of the system are irrelevant near the phase transition and the properties of one ladder are described by an Ising chain with transverse field. The mapping can be written explicitly by taking q each ”large junction” as a 1/2-spin with component 1 − φ2p (i) in the direction of the field. Near the phase transition, only the lowest non-zero order in the phase differences in ”large” junctions are relevant. Then, the first term in Eq. (1) represents the field contribution and the second the Ising coupling between nearby spins. The junctions located at the ends of the chains, blue crosses in Fig. (1d), couple the three ladders: n Uc = EJE cos [φ1 (1) + φ2 (1) + φ3 (1)] o + cos [φ1 (L) + φ2 (L) + φ3 (L)] , (2) where L is the double of the number of squares in each ladder. This contribution gives an extra coupling between spins at the boundaries in the Ising model. Effective model. Near the phase transition, the Hamiltonian of the three ladders of JJA maps to three Ising chains with transverse P magnetic field and coupling between their ends H = p Hp + Hc , where:

Josephson ladders Jc /J ∼ EJE /EJL . We are interested in the regime h/J ∼ 1 and Jc /J ∼ 1. A schematic representation of the Hamiltonian appears in Fig. (1d). In Eqs. (1,2), we have implicitly assumed that quantum fluctuations in the phase of the ”small” junctions are zero. In practice, these fluctuations are small but nonzero. We have also neglected any noise in the flux threading superconductor loops. These two contribution represent sources of incoherent noise for our effective Hamiltonian. We will comment later on how those contributions affect our results. We express the low-energy degrees of freedom for Hamiltonian Eq. (3,4) in terms of Majorana fermions. We use a multichannel version of the Jordan-Wigner transformation [9]. This mapping requires to enlarge the Hilbert space with an extra 1/2-spin. Operators acting on this spin are denoted by σ x (0), σ y (0) and σ z (0). The original spin operators are mapped to fermions via: c†p (k) = ηp (−1)

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